The **abstract prime factorization** (APF) or **prime signature** of a positive integer is the list of exponents in its prime factorization, conventionally written in descending order. The specific prime factors are ignored.

## Examples[]

- The number 88 has the prime factorization 2
^{3}× 11^{1}, and so an APF of [3, 1]. - The number 90 has the prime factorization 2
^{2}× 3^{3}, and so an APF of [3, 2].

## Numbers grouped listed by APF[]

All figures in dozenal. There are an infinite number of APF equivalence classes; the list below shows only those that include numbers less than or equal to 100.

APF | Numbers | OEIS ID | Description |
---|---|---|---|

[] | 1 | A000012 (?) (only the number 1) | a finite class consisting only of the number 1 |

[1] | 2, 3, 5, 7, E, 11, 15, ... | A000040 | prime numbers |

[2] | 4, 9, 21, 41, X1, 121, ... | A001248 | squares of primes |

[1, 1] | 6, X, 12, 13, 19, 1X, ... | A006881 | square-free semiprimes |

[3] | 8, 23, X5, 247, 92E, ... | A030078 | cubes of primes |

[2, 1] | 10, 16, 18, 24, 38, 39, ... | A054753 | squares of primes times a different prime |

[4] | 14, 69, 441, 1481, ... | A030514 | fourth powers of primes |

[3, 1] | 20, 34, 46, 48, 74, 88, ... | A065036 | cubes of primes times a different prime |

[1, 1, 1] | 26, 36, 56, 5X, 66, 86, ... | A007304 | products of three distinct primes (sphenic numbers) |

[5] | 28, 183, 1985, 9887, ... | A050997 | fifth powers of primes |

[2, 2] | 30, 84, 144, 169, 309, ... | A085986 | squares of squarefree semiprimes |

[4, 1] | 40, 68, 94, 116, 128, ... | A178739 | fourth powers of primes times a different prime |

[2, 1, 1] | 50, 70, 76, X6, E0, E8, ... | A085987 | square of a prime times two other distinct primes |

[6] | 54, 509, 9061, 58101, ... | A030516 | sixth powers of primes |

[3, 2] | 60, 90, 148, 288, 358, ... | A143610 | cubes of primes times squares of a different prime |

[5, 1] | 80, 114, 168, 254, ... | A178740 | fifth powers of primes times a different prime |

[3, 1, 1] | X0, 120, 1X0, 1E4, ... | A189975 | cube of a prime times two other distinct primes |

[7] | X8, 1323, 39265, ... | A092759 | seventh powers of primes |

[4, 2] | 100, 230, 294, 554, ... | A189988 | fourth powers of primes times squares of a different prime |